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On the Relationship between Minimum Norm and Linear Prediction for Spatial Spectrum Estimation Sverre Holm Department of Informatics, University of Oslo 22 April 1998 Abstract array. This gives a special form which consists of finding the roots of a polynomial; the root-minimum norm method. The method was generalized to an arbitrary array geometry in [3]. In its general form the minimum norm method consists of finding the steering vector that lies in the noise subspace, where the first element is unity, and where the length of the vector is minimized. It is an alternative to the MUSIC method [4] in that this method finds the steering vector that is closest to the noise subspace. The minimum norm method for spatial spectrum estimation is derived from the linear prediction method in exactly the same way as the MUSIC method is derived from the minimum variance method. The derivation consists of replacing the correlation with its noise subspace component and setting all noise eigenvalues to unity. This makes it simpler to understand the methods and their properties. This relationship also brings out the meaning of setting the first element to unity in the minimum norm method — it corresponds to the predicted element in linear prediction. There is also a parallel between properties: e.g. just as linear prediction has a lower detection threshold than minimum variance so does minimum norm compared to MUSIC. Thus the properties of the subspace methods seem to be ’inherited’ from the original non-subspace methods. In this correspondence we will exploit the relationship between the minimum norm method and the linear prediction method in order to give a derivation of the method that fits into the framework of the spatial spectrum estimation presentation in sections 7.27.3 of [5]. We will also show how this explains some of the properties of the method. 1 Introduction 2 The minimum norm method for spatial spectrum estimation is a subspace-based method that was first presented completely by Kumaresan and Tufts [1]. Some of the ideas had already been given by Reddi [2]. These first references both use a uniform linear The derivation is based on [5] and uses the same terminology. Consider an array of M sensors at arbitrary locations with a spatial correlation matrix R, ~ and a steering vector from direction ζ~ which is e(ζ). 1 Theory 2.1 Minimum Variance and MUSIC Another interesting link between the minimum variance and MUSIC methods is found by considThe minimum variance method is derived from the ering estimators that are nonlinear functions of the consideration that the output energy of the beamcovariance matrix [6]: former should be minimized while the response to the actual direction given by ζ~ is unity. This gives ~ = e0 (ζ)R ~ −k e(ζ). ~ fk (ζ) (5) the following ouput power: The minimum variance estimator is found by letting ~ = [e0 (ζ)R ~ −1 e(ζ)] ~ −1 P M V (ζ) (1) k = 1: ~ = f1 (ζ) ~ −1 . P M V (ζ) (6) where e0 means the conjugate transpose of e. In all the subsequent subspace methods a model consisting When k grows the larger eigenvalues are more and of Ns incoherent plane waves in additive white noise more neglected and in the limit the MUSIC estimate is assumed. The spatial correlation matrix can then is obtained: be divided into a rank Ns signal plus noise subspace ~ = lim [σ 2 k · fk (ζ)] ~ −1 . and a rank M − Ns noise subspace: P M U SIC (ζ) (7) k→∞ R= Ns X i=1 λi vi vi0 + M X λi vi vi0 . (2) 2.2 Linear Prediction and Minimum Norm i=Ns +1 The linear prediction (LP) method is a signal modeling approach where the output of sensor m0 is assumed to be a weighted linear combination of the outputs of the other sensors, thus a model order of M is assumed. The method is derived from the consideration that the output energy of the beamformer should be minimized while the weight vector for el0 0 R = Vs Λs Vs + Vn Λn Vn (3) ement m0 should be unity. This gives the following ouput power: where the subscripts stand for signal-plus-noise sub0 R−1 δ δm m0 0 space and noise subspace respectively. ~ = P LP (ζ) . (8) 0 −1 0 ~ ~ [e(ζ) R δm0 δm0 R−1 e(ζ)] The MUSIC method prescribes that the spatial spectrum estimate is found as the steering vector which is closest to the noise subspace. Thus the norm Here δm0 is an M-element column vector with all ~ n V 0 e(ζ) ~ should be minimized. This can also zeros except for a 1 in position m0 . In order to sime0 (ζ)V n be interpreted as replacing the correlation matrix in plify the comparison, we will neglect the numerator, (1) with the noise subspace component from (3), and because it is invariant with direction. For a linear at the same time performing a whitening of the spa- array, the predicted element m0 should be selected tial noise spectrum, i.e. setting all noise eigenvalues at one of the ends. Positions in the center of the array usually give reduced resolution [7]. A causal to unity. Thus the MUSIC estimate is: model, as used in time series analysis, would preM U SIC ~ 0 ~ 0 ~ −1 P (ζ) = [e (ζ)Vn Vn e(ζ)] . (4) scribe m0 = M . For arbitrary geometries, ends can It is assumed that the eigenvalues have been sorted, i.e. that λ1 is the largest and λM is the smallest eigenvalue. For a white background noise σ 2 = λi for i = Ns+1 , . . . , M , where σ 2 is the noise power. The spectral decomposition can be written in a more compact form as: 2 Yu [8] develop a total least squares solution to the linear prediction problem which is in fact the minimum norm solution [9]. However, the close parallel between minimum variance and MUSIC on the one hand, and linear prediction and minimum norm on the other hand, has not been fully exploited before. usually not be defined, so an element near the phase center should be chosen. Now we apply the same subspace idea as in the MUSIC method, i.e. replace the correlation matrix in the numerator of (8) with the noise subspace component from (3), and perform a whitening of the spatial noise spectrum. This gives: ~ = [e(ζ) ~ 0 Vn V 0 δm δ0 Vn V 0 e(ζ)] ~ −1 . (9) 2.3 P M N (ζ) 0 m0 n n This equation gives a spatial spectrum estimate which consists of finding the steering vector that lies in the noise subspace, where the element m0 is unity and the length of the vector is required to be minimum. Thus it is a generalization of the minimum norm algorithm. The relationship with the linear prediction method also gives a basis for the selection of the seemingly arbitrary value of m0 = 1 in the minimum norm method. The relationship between linear prediction and minimum norm can also be found by considering linear prediction type estimators that are nonlinear functions of the covariance matrix: ~ = e(ζ) ~ 0 R−k δm δ0 R−k e(ζ). ~ gk (ζ) 0 m0 A natural extension to the methods presented here is to skip the prewhitening, i.e. let the noise eigenvalues remain in the expressions. This results in a variation of the MUSIC method called the eigenvector (EV) method [7]: ~ = [e0 (ζ)V ~ n Λ−1 V 0 e(ζ)] ~ −1 . P EV (ζ) n n (10) In the limit as k grows, the minimum norm estimate is obtained: ~ = lim [σ P M N (ζ) k→∞ ~ −1 . · gk (ζ)] (13) This method has an advantage in scenarios with nonwhite background noise. It is also more robust to incorrect estimation of the number of signals, Ns , than the MUSIC method. This can e.g. be seen from the fact that the method degenerates to the minimum variance method when Ns = 0, while the MUSIC method will give a constant, independent of direction. Based on the same principle one can develop a method from the ideas of linear prediction and minimum norm which we call the LP-EV method or the weighted minimum norm method. The linear prediction estimator is found by letting k = 1: ~ = g1 (ζ) ~ −1 . P LP (ζ) (11) 2k Generalization to Eigenvector and LPEV methods ~ = [e(ζ) ~ 0 Vn Λ−1 V 0 δm δ0 Vn Λ−1 V 0 e(ζ)] ~ −1 . P LP −EV (ζ) n n 0 m0 n n (12) This method consists of finding the minimum norm steering vector in the weighted noise subspace when element m0 is constrained to be unity. The weighting consists of normalizing each noise eigenvector according to its contribution, i.e. by the inverse of the eigenvalue. This method shares the same properties as the eigenvector method with respect to non-white backgrounds, it has reduced sensitivity to The relationship given here between linear prediction and minimum norm is not a totally new one. In [1], Kumaresan and Tufts use the similarity with the autocorrelation method of linear prediction in their derivation. In [3] the relationship with linear prediction is used to find a solution called the minimum norm linear prediction vector. Further Rahman and 3 incorrect estimation of the number of signals, and degenerates to the linear prediction method when Ns = 0. This method was actually first developed by Johnson [10], but its relationship to the minimum norm method has not been noted before. the linear prediction and minimum variance methods. As k goes to infinity, the minimum norm and MUSIC methods are obtained. This parallel development has not been exploited before. We believe it will benefit the presentation of these methods. It also gives a rationale for the seemingly arbitrary choice of forcing the first element to be unity in the minimum norm method; this element is now seen to be equivalent to the predicted element in the linear prediction method. We have also observed that properties such as resolution, detection threshold, and variance seem to be ’inherited’ from the non-subspace methods, e.g. just as linear prediction has a lower detection threshold than the minimum variance method so does minimum norm have a lower threshold than MUSIC. 3 Comparison of Properties The linear prediction method has improved resolution compared to minimum variance. It also has a lower detection threshold. Its disadvantage is a larger background fluctuation in non-signal regions [5]. Similarly, the minimum norm algorithm has a smaller bias and a lower detection threshold than the MUSIC method [11]. According to [12], these two properties follow each other. In [13] it is shown that for small samples, minimum norm has larger variance than MUSIC. It has also been shown that of all the MUSIC-like algorithms, the original MUSIC algorithm has the smallest large-sample variance [14] compared to e.g. minimum-norm. Thus it is seen that to some extent comparison between LP and min-norm, and minimum variance and MUSIC bring out almost parallel properties: Less bias and lower detection threshold for linear prediction and minimum norm, and smaller variance for minimum variance and MUSIC. References [1] R. Kumaresan and D. W. Tufts, “Estimating the angle of arrival of multiple plane waves,” IEEE Trans. Aeorospace and Electr. Syst, vol. AES19, pp. 134–139, Jan. 1983. [2] S. S. Reddi, “Multiple source location - a digital approach,” IEEE Trans. Aeorospace and Electr. Syst, vol. AES-15, pp. 95–105, Jan. 1979. 4 Conclusion [3] F. Li, R. J. Vaccaro, and D. W. Tufts, “Minnorm linear prediction for arbitrary sensor arrays,” in IEEE Int. Conf. Acoust., Speech, Sign. Proc., pp. 2613–2616, 1989. The minimum norm method has been derived from the linear prediction method for spatial spectrum estimation. The derivation is exactly parallel to the derivation of MUSIC from the minimum variance method. The derivation consists of replacing the correlation with its noise subspace component and setting all noise eigenvalues to unity. The parallel derivations can also be seen by considering the methods as functions depending on R−k ; k = 1 gives [4] R. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proc. of the RADC Spectrum Estimation Workshop, pp. 243–258, 1979. 4 MUSIC, MIN-NORM, and FINE,” IEEE Trans. Signal Processing, vol. 42, pp. 1812– 1816, July 1994. [5] D. H. Johnson and D. E. Dudgeon, Array Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1993. [6] C. Vaidyanathan and K. M. Buckley, “Perfor- [14] P. Stoica and A. Nehorai, “MUSIC, maximum mance analysis of DOA estimation based on likelihood, and Cramer-Rao bound: Further nonlinear functions of covariance matrix,” Sigresults and comparisons,” IEEE Trans. Signal nal Processing, vol. 50, pp. 5–16, 1996. Processing, vol. 38, pp. 2140–2150, Dec. 1990. [7] D. H. Johnson and S. 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Kaveh, “Comparative study of the biases of MUSIC-like estimators,” Signal Processing, vol. 50, pp. 39–55, 1996. [13] X.-L. Xu and K. M. Buckley, “Bias and variance of direction-of-arrival estimates from 5